The Pancharatnam-Berry phase is a geometrical phase associated with the polarization of light. When the polarization of a beam traverses a closed loop on the Poincare sphere, the final state differs from the initial state by a phase factor equal to half the area Ω, encompassed by the loop on the sphere (see S. Pancharatnam, Proc. Ind. Acad. Sci. A 44, 247 (1956), and M. V. Berry, Proc. Roy. Soc. (London) A 392, 45 (1984)). In a typical experiment, the polarization of a uniformly polarized beam is altered by a series of space-invariant (transversely homogenous) waveplates and polarizers, and the phase, which evolves in the time-domain is measured by means of interference (see R. Simon, H. J. Kimble and E. C. G. Sudharshan, Phys. Rev. Lett. 61, 19 (1988), and P. G. Kwiat and R. Y. Chiao, Phys. Rev. Lett. 66, 588 (1991)).
Recently, a Pancharatnam-Berry phase in the space-domain was considered. Using space-variant (transversely inhomogeneous) metal stripe subwavelength gratings, we demonstrated conversion of circular polarization into radial polarization (see Z. Bomzon, V. Kleiner and E. Hasman, Appl. Phys. Lett. 79, 1587 (2001)), and showed that the conversion was accompanied by a space-variant phase modification of geometrical origin that effected the propagation of the beams (Z. Bomzon, V. Kleiner, and E. Hasman, Opt. Lett. 26, 1424 (2001)).
Previously Bhandari suggested using a discontinuous spatially varying waveplate as a lens based on similar geometrical phase effects (R. Bhandari, Phys. Rep. 281, 1 (1997)). Recent studies have investigated periodic polarization gratings (F. Gori, Opt. Lett. 24, 584 (1999), C. R. Fernández-Pousa, I. Moreno, J. A. Davis and J. Adachi, Opt. Lett. 26, 1651 (2001) and J. Tervo, and J. Turunen, Opt. Lett. 25, 785 (2000)). These authors showed that the polarization of the diffracted orders could differ from the polarization of the incident beam. We intend to prove and utilize a connection between the properties of such polarization gratings and the space-domain Pancharatnam-Berry phase.